# Transporting a Cauchy foliation of Minkowski space

Consider a spacetime $$(\zeta^{3,1},g)$$

where $$g=\frac{du\,dv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2} \quad \forall u,v,r,w \in (0,1)$$ Now this is just Minkowski space in different coordinates (related to Dirac/Light cone coordinates/null coordinates). I'm looking to take a Cauchy foliation of $$\zeta^{3,1},$$ change the metric back to $$g'=dudv-dw^2-dr^2,$$ and use the induced measure from $$g'$$ by means of the volume form to transform the foliated past light cone region onto a new spacetime $$(\Psi^{~3,1},d).$$

Let's look at the steps involved for the $$(1+1)$$ dimensional case $$(\zeta^{1,1},u)$$ where $$u=\frac{dxdq}{xq}.$$ The Cauchy foliation is simply $$\ln(b)\ln(y)=t$$ (solving for $$y$$ gives an explicit representation) which can then be transformed via Mellin-like transform:

$$\Phi_s(t)= \int_{S=(0,1)} \exp {\frac{t s}{\ln b}}~db = \int_0^1 tb^{t-1} \exp \frac{s}{\ln b}~db = 2\sqrt{ts}K_1(2\sqrt{ts})$$

Observe that this is an unnormalized K-distribution and it's inverse transform yields an unnormalized distribution call it the "$$\zeta$$-distribution" (our Cauchy foliation). Observe that the Fisher information metric of the $$\zeta$$-distribution is $$\Phi_s(t)$$ up to a factor. Observe that the $$\zeta$$-distribution yields solutions to the Killing field $$X=\langle x \ln x, -y\ln y \rangle.$$ It also provides a particular distributional solution to the following diffusion equation with diffusivity depending on both space and time:

$$\frac{\partial^2}{\partial t^2}\nu(t,b)=-\frac{b}{t}\frac{\partial}{\partial b}\nu(t,b)$$

In short, $$\Phi_1(t)=v$$ is a Lorentzian metric on the smooth $$(1+0)$$ manifold $$(\Psi^{~1,0}, v).$$ We have that $$\Psi^{~3,1}=\Psi^{~1,1}\times \Psi^{~1,0} \times \Psi^{~1,0}$$ so we can define the product metric on $$\Psi^{~3,1}.$$

There are two different constructions going on here. The first construction starts with a bonafide spacetime and transforms the past light cone region onto a new spacetime. The second construction begins by simplifying the spacetime down to $$(1+1)$$ dimensions, transforms the past light cone region, and then scales that manifold with the Cartesian product, furnishing with the product metric.

Does restricting to $$g'$$ and transforming the foliated past light cone preserve all or most of the information contained in $$(\zeta^{3,1},g)?$$ Is $$(\Psi^{~3,1},d)$$ also a spacetime, or a component of one?

We are given a spacetime: $$(\zeta,g)$$ which we recognize as Minkowski space in different coordinates.

We start by observing that the product space of the foliations:

$$\Omega_{t,r,\theta}(x,y,z)= \varphi_t(x)\varphi_r(y)\varphi_{\theta}(z) \space\space\space\space\space{x,y,z\in(0,1)} \space\space\space{t,r,\theta >0}$$

Noticeably yields a product space $$\Bbb R^{3,3}$$ where $$(t,r,\theta)$$ are time variables and $$(x,y,z)$$ are space variables. This $$6$$-manifold is not entirely helpful, so we reduce to $$(3+1)$$ dimensions by letting $$s=t=r=\theta.$$ This collapses the time variables down to just one, which is what we want.

Now we are left with a Cauchy foliation, $$\Omega$$, of a $$(3+1)$$ spacetime, in fact this spacetime is precisely the $$\zeta$$ we were initially given:

$$\Omega_s(x,y,z)=\varphi_s(x)\varphi_s(y)\varphi_s(z)\space\space\space\space{s>0}$$

And now we restrict the metric to $$h=dudv-dr^2-dw^2$$ and hit each leaf with the transform:

$$\int_{(0,1)^3} \Omega_s(x,z,y)~dxdydz=\bigg(\int_{(0,1)} \Omega_s(x)~dx\bigg)^3= \psi^3(s)$$

We have an explicit function to describe the leaves of the Cauchy foliation and we now have an explicit function for the hyper-volume. This transform maps from the spatial domain to the time domain. Whether that has any physical relevance, I do not know.

We can generalize the integral transform to:

$$\int_{(0,1)^3} x^{a}y^{b}z^{c}\Omega_s(x,z,y)~\frac{dxdydz}{xyz}=\chi_s(a,b,c).$$

And so we have a mapping given by the integral transform:

$$T: \Omega \to \chi$$

We have that $$T$$ is linear because it is an integral transform on bounded domain, which we know to be linear. And as $$s$$ varies continuously on $$\Omega$$ the corresponding change of $$s$$ is also continuous on $$\chi$$. The map is smooth as well, because a smooth change in $$s$$ on $$\Omega$$ corresponds to a smooth change of $$s$$ on $$\chi$$.

Therefore the problem is reduced to finding a complete Lorentzian metric that is preserved by an infinitesimally generated flow on $$\chi_s(a,b,c).$$ We are looking for a Killing field. Particularly of interest are the cases $$\lim_{s \to 0}\chi_s$$ and $$\lim_{s \to \infty} \chi_s$$ which serve to define boundaries on the resultant spacetime. The spacetime in question is therfore non-compact with boundary.